Solutions of Local and Nonlocal Discrete Complex Modified Korteweg–De Vries Equations and Continuum Limits

Abstract

Cauchy matrix approach for the discrete Ablowitz–Kaup–Newell–Segur equations is reconsidered, where two “proper” discrete Ablowitz–Kaup–Newell–Segur equations and two “unproper” discrete Ablowitz–Kaup–Newell–Segur equations are derived. The “proper” equations admit local reduction, while the “unproper” equations admit nonlocal reduction. By imposing the local and nonlocal complex reductions on the obtained discrete Ablowitz–Kaup–Newell–Segur equations, two local and nonlocal discrete complex modified Korteweg–de Vries equations are constructed. For the obtained local and nonlocal discrete complex modified Korteweg–de Vries equations, soliton solutions and Jordan‐block solutions are presented by solving the determining equation set. The dynamical behaviors of 1‐soliton solution are analyzed and illustrated. Continuum limits of the resulting local and nonlocal discrete complex modified Korteweg–de Vries equations are discussed.